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Highly Connected Sets and the Excluded Grid Theorem

1999
*
Journal of combinatorial theory. Series B (Print)
*

We present a short proof of

doi:10.1006/jctb.1998.1862
fatcat:urjg4vgcq5ep3nlxaq36ekfweu
*the**excluded**grid**theorem*of Robertson*and*Seymour,*the*fact that a graph has no large*grid*minor if*and*only if it has small tree-width. ... We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if*and*only if it contains no large*highly**connected**set*of vertices. ...*Highly**connected**sets**and*tangles*The*proofs of*Theorem*2 given in 7]*and*3] rely heavily on a concept central to*the*Robertson-Seymour theory of minors but not so far considered in this paper,*the*concept ...##
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Contraction Bidimensionality: The Accurate Picture
[chapter]

2009
*
Lecture Notes in Computer Science
*

We provide new combinatorial

doi:10.1007/978-3-642-04128-0_63
fatcat:viiyykcgtzf5lbiurgekwtwesa
*theorems*on*the*structure of graphs that are contained as contractions in graphs of large treewidth. ... As a consequence of our combinatorial results we unify*and*significantly simplify contraction bidimensionality theory-*the*meta algorithmic framework to design efficient parameterized*and*approximation ... Finally, we show*the*following analogue of*the**Excluding**Grid*-minor*Theorem*. ...##
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TOWARDS A MATROID-MINOR STRUCTURE THEORY
[chapter]

2007
*
Combinatorics, Complexity, and Chance
*

In combination with

doi:10.1093/acprof:oso/9780198571278.003.0005
fatcat:rnrqxwsn6fdhdixknsck3bxqqy
*the**grid**theorem*this yields*the*following*theorem*. ...*Theorem*5. 5 ( 5*Grid**theorem*for matroids) Let H be a planar graph*and*q be a prime power. ...##
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The Structure of W_4-Immersion-Free Graphs
[article]

2016
*
arXiv
*
pre-print

We study

arXiv:1602.02002v1
fatcat:ewc656g3grgtpbpkn3oedndpki
*the*structure of graphs that do not contain*the*wheel on 5 vertices W4 as an immersion,*and*show that these graphs can be constructed via 1, 2,*and*3-edge-sums from subcubic graphs*and*graphs ... Lemma 2 essentially states that large treewidth yields a large number of vertex disjoint cycles that are*highly**connected*to each other,*and*an additional disjoint*set*that is*highly**connected*to these ... Other major results in graph minor theory include*the*(Strong) Structure*Theorem*[28] ,*the*Weak Structure*Theorem*[27] ,*the**Excluded**Grid**Theorem*[26, 31, 21] , as well as numerous others, e.g., ...##
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Some Recent Progress and Applications in Graph Minor Theory

2007
*
Graphs and Combinatorics
*

In

doi:10.1007/s00373-006-0684-x
fatcat:wkf3w6cemzc4hnnw3neqpck6hu
*the*core of*the*seminal Graph Minor Theory of Robertson*and*Seymour lies a powerful*theorem*capturing*the*"rough" structure of graphs*excluding*a fixed minor. ... This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under*the*graph minor relation. ...*Theorems*5.3*and*5.4 are sharp in*the*sense that*the*7-*connectivity**and*9-*connectivity*(respectively) conditions cannot be relaxed, as mentioned above. ...##
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Page 46 of Mathematical Reviews Vol. , Issue 2000a
[page]

2000
*
Mathematical Reviews
*

(RS-INT2; Moscow); Thomassen, Carsten (DK-TUD; Lyngby)

*Highly**connected**sets**and**the**excluded**grid**theorem*. (English summary) J. Combin. Theory Ser. B 75 (1999), no. 1, 61-73. ... Summary: “We present a short proof of*the**excluded**grid**theorem*of Robertson*and*Seymour,*the*fact that a graph has no large*grid*minor if*and*only if it has small tree-width. ...##
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On the block number of graphs
[article]

2017
*
arXiv
*
pre-print

We also study k-blocks in graphs from classes of graphs G that

arXiv:1702.04245v1
fatcat:n56svml2mzgftji5es3rqhuv3u
*exclude*some fixed graph as a topological minor,*and*prove that every G ∈G satisfies β(G) ≤ c√(|G|) for some constant c = c( G). ...*The*block number β(G) of G is*the*maximum integer k for which G contains a k-block. ... of*Theorem*3. ...##
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Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)

2014
*
Symposium on Theoretical Aspects of Computer Science
*

On

doi:10.4230/lipics.stacs.2014.542
dblp:conf/stacs/MarxP14
fatcat:dy4trfjwkvhj3i2tzcvrguedyi
*the*algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus,*and*feedback vertex*set*number of G gives rise to a*highly*nontrivial algorithm for Subgraph ... We show that all*the*questions arising in this framework are answered by a*set*of 11 maximal positive results (algorithms)*and*a*set*of 17 maximal negative results (hardness proofs); some of these results ... As*Theorems*2*and*3 show, |V (H)| is a*highly*relevant parameter for*the*problem. ...##
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Graph Minors and Parameterized Algorithm Design
[chapter]

2012
*
Lecture Notes in Computer Science
*

We discuss its direct meta-algorithmic consequences, we present

doi:10.1007/978-3-642-30891-8_13
fatcat:56shhf7tevfdpczn4m7qir3zti
*the*algorithmic applications of core*theorems*such as*the**grid*-exclusion*theorem*,*and*we give a brief description of*the*irrelevant vertex ... We present some of*the*basic algorithmic techniques*and*methods that emerged from this theory. ... Acknowledgements We wish to thank Isolde Adler, Marcin Kamiński*and*Stavros G. Kolliopoulos for their detailed comments, remarks,*and*suggestions on this text. ...##
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Layered separators in minor-closed graph classes with applications

2017
*
Journal of combinatorial theory. Series B (Print)
*

Our results imply that every n-vertex graph

doi:10.1016/j.jctb.2017.05.006
fatcat:gzpqanrjr5ebddcetqpoaa6rvm
*excluding*a fixed minor has a 3-dimensional*grid*drawing with n^O(1)n volume, whereas*the*previous best bound was O(n^3/2). ... We extend these results with a O( n) bound on*the*nonrepetitive chromatic number of graphs*excluding*a fixed topological minor,*and*a ^O(1)n bound on*the*queue-number of graphs*excluding*a fixed minor. ... Thanks to Zdeněk Dvořák, Gwenaël Joret, Sergey Norin, Bruce Reed*and*Paul Seymour for helpful discussions. Thanks to*the*anonymous referees for numerous helpful comments. ...##
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The Complexity of Conjunctive Queries with Degree 2
[article]

2022
*
arXiv
*
pre-print

Using dilutions we observe an analogue to

arXiv:2111.11532v3
fatcat:lxuylqth3zfgjfpf2lex4szuha
*the**Excluded**Grid**Theorem*for degree 2 hypergraphs. ... We also generalise our main structural result to arbitrary bounded degree*and*discuss possible paths towards a characterisation of*the*bounded degree case. ... Grohe's lower bound critically relies on*the**Excluded**Grid**Theorem*by Robertson*and*Seymour [27] . ...##
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Distribution grid topology reconstruction: An information theoretic approach

2015
*
2015 North American Power Symposium (NAPS)
*

One of

doi:10.1109/naps.2015.7335248
fatcat:gcyiipid6vfrxl7wxqmmwh5oc4
*the*key issues is frequent distribution*grid*re-configuration, which is hard to detect based on traditional approaches. ... Wrong topology information causes wrong control signal, making fast changing smart*grid*prone to go over stability boundaries*and*to collapse. ... ACKNOWLEDGMENT*The*second author would like to thank Tie Liu from Texas A&M University for discussions on*the*Chow-Liu algorithm. ...##
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Recent work in matroid representation theory

2005
*
Discrete Mathematics
*

This paper surveys recent work in matroid representation theory

doi:10.1016/j.disc.2004.07.039
fatcat:it6qr2fri5gntjitdyoa7fk43m
*and*discusses a number of open problems. ...*Sets*of fields:*excluded*minors*The*deepest part of*Theorem*3.1 is*the**excluded*-minor characterization of regular matroids. ... Firstly there are no analogues to*the*Wheels*and*Whirls*Theorem**and**the*Splitter*Theorem*for vertically 4-*connected*matroids. ...##
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Extremal infinite graph theory

2011
*
Discrete Mathematics
*

We survey various aspects of infinite extremal graph theory

doi:10.1016/j.disc.2010.12.018
fatcat:4enetufd25ddbhke2ga4qham2a
*and*prove several new results.*The*lead role play*the*parameters*connectivity**and*degree. This includes*the*end degree. ... Acknowledgement*The*author was supported by Fondecyt grant no. 11090141. ... We use*the*vertex-/edge-degree of*the*ends (for a definition see below) to force*highly**connected*subgraphs*and**grid*minors. ...##
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On the tree-width of even-hole-free graphs
[article]

2020
*
arXiv
*
pre-print

This can be seen as a strengthening of Robertson

arXiv:2008.05504v1
fatcat:ij2smflzl5dx5h6sladobes5ve
*and*Seymour's*excluded**grid**theorem*for*the*case of minor-free graphs. ... Our*theorem*implies that every class of even-hole-free graphs*excluding*a fixed graph as a minor has bounded tree-width. ... Even-hole-free graphs*excluding*a minor In this section we prove an 'induced*grid**theorem*' for graphs*excluding*a fixed minor. ...
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